hat Expected Utility The consistency requirements include reflexivity, completeness and transitivity as well as two others. A lottery r is a set of possible outcomes rI, .. In) and a set of probabilities for each outcome ip The representation theorem states that a preference relation over lotteries, > that satisfies these conditions can be represented by a ton-Neumann Morgenstern expected utility function > y÷→ pu(x)2∑q qiu(yi) The vNM expected utility for lottery r is U(r)=iipa u(ri)where u(ai)is the regular utility from outcome I, In the earlier example, suppose the utility from each outcome is simply the money payoff. The probability of each utcome is known. If the consumers preferences can be represented by such a VNM expected utility function then: U(c)=(0.8×1000+(0.2×0)andU(n)=(1×50)soU(c)>U(n) thus c>n In this example the consumer prefers the lottery c and will buy the car Market- What Nerty Risk aversion The consumer has taken the action involving uncertainty. Do they then like risk? Not necessarily A common assumption is that of risk aversion. A consumer is risk averse when their utility function is concave. Suppose there is just one good- money m. The consumer gets regular utility u(m) from money m (5) u(1)+u(9) u(1) In the above diagram the consumer prefers 5 for sure than 9 with probability half and I with probability half. The expected value of the latter lottery is 5. But u(5)2 u(1)+=u(9). This is because u is concave- risk aversion the earlier This is a case ofMarket — What Next? 3 Expected Utility • The consistency requirements include reflexivity, completeness and transitivity as well as two others. • A lottery x is a set of possible outcomes {x1, . . . , xn} and a set of probabilities for each outcome {p1, . . . , pn}. • The representation theorem states that a preference relation over lotteries, º, that satisfies these conditions can be represented by a von-Neumann Morgernstern expected utility function: x º y ⇐⇒ Xn i=1 piu(xi) ≥ Xm i=1 qiu(yi) • The vNM expected utility for lottery x is U(x) = Pn i=1 piu(xi) where u(xi) is the regular utility from outcome xi . • In the earlier example, suppose the utility from each outcome is simply the money payoff. The probability of each outcome is known. If the consumers preferences can be represented by such a vNM expected utility function then: U(c) = (0.8 × 1000) + (0.2 × 0) and U(n) = (1 × 500) so U(c) > U(n) thus c º n • In this example the consumer prefers the lottery c and will buy the car. Market — What Next? 4 Risk Aversion • The consumer has taken the action involving uncertainty. Do they then like risk? Not necessarily. • A common assumption is that of risk aversion. A consumer is risk averse when their utility function is concave. Suppose there is just one good — money m. The consumer gets regular utility u(m) from money m. . . . . . . . . . . ............. ... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. . . . . . . ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ....................................................................................................................................................................................................................................................................................................................................................................................................................... .............................................................................................................................................................................................. . ................................................................................................................................................................................................................................................................................................................................................ 0 u m u(m) • • • • 1 5 9 u(1) u(5) u(9) 1 2 u(1) + 1 2 u(9) • In the above diagram the consumer prefers 5 for sure than 9 with probability half and 1 with probability half. The expected value of the latter lottery is 5. But u(5) ≥ 1 2 u(1) + 1 2 u(9). This is because u is concave — risk aversion. • In the earlier example the consumer had u(m) = m. This is a case of risk neutrality